zbMATH — the first resource for mathematics

$$\mathcal D$$-bundles and integrable hierarchies. (English) Zbl 1253.14020
This article is on the geometry of $$\mathcal{D}$$-bundle, i.e., locally projective $$\mathcal{D}$$-modules, on algebraic curves, and applications of them to the studies of integrable hierarchies, i.e., the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies, and especially to the rational, trigonometric, and elliptic solutions of KP.
Since the local structure of the $$\mathcal{D}$$-bundles is determined by the full Sato Grassmannian, the authors show that the KP hierarchies have a geometric description as flows on moduli space of the $$\mathcal{D}$$-bundles. These solutions are captured by the $$\mathcal{D}$$-bundles on cubic curves $$E$$, i.e., irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1.
Fourier-Mukai transform describing the $$\mathcal{D}$$-modules on the cubic curves $$E$$ in terms of (complexes of) sheaves on a twisted cotangent bundle $$E^\natural$$ over $$E$$ shows the correspondence between the $$\mathcal{D}$$-bundles and the CM spectral sheaves.
It implies that the poles of the KP-solutions are identified with the positions of the CM particles. They also discuss their relations to the Hitchin integrable system.

MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14H20 Singularities of curves, local rings 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 14H52 Elliptic curves 17B65 Infinite-dimensional Lie (super)algebras
Full Text:
References:
 [1] Airault, H., McKean, H. P., Moser, J.: Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30, 95-148 (1977) · Zbl 0338.35024 · doi:10.1002/cpa.3160300106 [2] Akhmetshin, A., Krichever, I., Volvovski\?ı, Y.: Elliptic families of solutions of the Kadomtsev-Petviashvili equations and the field elliptic Calogero-Moser system. Funk- tsional. Anal. i Prilozhen. 36, no. 4, 1-17 (2002) (in Russian) · Zbl 1033.37032 · doi:10.1023/A:1021706525301 [3] Álvarez, A., Muñoz, J., Plaza, F.: The algebraic formalism of soliton equations over arbi- trary base fields. In: Workshop on Abelian Varieties and Theta Functions (Morelia, 1996), Aportaciones Mat. Investig. 13, Soc. Mat. Mexicana, 3-40 (1998) · Zbl 0995.14021 [4] Arbarello, E.: Sketches of KdV. In: Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), Contemp. Math. 312, Amer. Math. Soc., 9-69 (2002) · Zbl 1056.14023 [5] Asensio, M., Van den Bergh, M., Van Oystaeyen, F.: A new algebraic approach to microlocalization of filtered rings. Trans. Amer. Math. Soc. 316, 537-553 (1989) · Zbl 0686.16002 · doi:10.2307/2001360 [6] Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems. Cambridge Monogr. Math. Phys., Cambridge Univ. Press (2003) · Zbl 1045.37033 · doi:10.1017/CBO9780511535024 [7] Babelon, O., Billey, E., Krichever, I., Talon, M.: Spin generalization of the Calogero- Moser system and the matrix KP equation. In: Topics in Topology and Mathemat- ical Physics, Amer. Math. Soc. Transl. (2) 170, Amer. Math. Soc., 83-119 (1995) · Zbl 0843.58069 [8] Baranovsky, V., Ginzburg, V., Kuznetsov, A.: Quiver varieties and a noncommutative 2 P . Compos. Math. 134, 283-318 (2002) · Zbl 1048.14001 · doi:10.1023/A:1020930501291 [9] Baranovsky, V., Ginzburg, V., Kuznetsov, A.: Wilson’s Grassmannian and a noncom- mutative quadric, Int. Math. Res. Notices 2003, no. 21, 1155-1197 · Zbl 1059.58006 · doi:10.1155/S1073792803210126 · arxiv:math/0203116 [10] Beilinson, A., Bernstein, J.: Proof of Jantzen conjectures. In: I. M. Gelfand Seminar, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc., 1-50 (1993) · Zbl 0790.22007 [11] Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigen- sheaves. · Zbl 0864.14007 · www.math.uchicago.edu [12] Beilinson, A., Drinfeld, V.: Chiral Algebras. Amer. Math. Soc. Colloq. Publ. 51, Amer. Math. Soc. (2004) · Zbl 1138.17300 [13] Beilinson, A., Drinfeld, V.: Opers. arXiv:math.AG/0501398 [14] Beilinson, A., Schechtman, V.: Determinant bundles and Virasoro algebras. Comm. Math. Phys. 118, 651-701 (1988) · Zbl 0665.17010 · doi:10.1007/BF01221114
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.